Group #32973006 — subgroups

Subgroups — #32973006

Order 18688.

212 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22611730	C₂	1	1	yes	maximal, characteristic
2	C₈×#11924012	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#2150	C₂	1	1	yes	maximal, characteristic
4	#17988640	C₄	1	1	yes	characteristic
5	C₄×#11924012	C₄	1	1	yes	characteristic
6	C₄⋊#11924003	—	2	1	no	—
7	C₂²⋊#11924003	C₄	1	2	yes	—
8	C₄×#11924003	—	2	1	no	—
9	#17988309	—	2	1	no	—
10	C₈×#10596831	—	2	1	no	—
11	C₂×M₃₂×C₇₃	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₇₃	—	2	1	no	—
14	C₄⋊C₁₆×C₇₃	—	2	1	no	—
15	C₂×#11924012	—	2	1	no	—
16	C₂×#11924003	—	2	2	no	—
17	C₄⋊#10596831	C₈	1	2	yes	—
18	C₂²⋊#10596831	—	2	1	no	—
19	#14556502	C₈	1	2	yes	—
20	C₄×#10596831	—	2	1	no	—
21	C₄×#10596831	—	2	1	no	—
22	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
25	M₃₂×C₇₃	—	4	2	no	—
26	C₂×C₁₆×C₇₃	—	2	2	no	—
27	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
31	#11924012	—	2	4	no	—
32	C₂×#10596831	—	2	2	no	—
33	C₂×#10596831	—	2	2	no	—
34	#11924003	—	4	2	no	—
35	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
38	C₂×C₈×C₇₃	—	2	1	no	—
39	C₂×C₈×C₇₃	—	2	1	no	—
40	C₂×C₈×C₇₃	—	4	1	no	—
41	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
48	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
49	C₁₆×C₇₃	—	4	2	no	—
50	C₂³×C₇₃	#97	1	1	yes	characteristic
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	—	2	2	no	—
54	C₂×C₄×C₇₃	—	2	2	no	—
55	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
61	C₈×C₇₃	—	2	1	no	—
62	C₈×C₇₃	—	2	1	no	—
63	C₈×C₇₃	—	2	1	no	—
64	C₈×C₇₃	#100	1	1	yes	characteristic
65	C₈×C₇₃	#101	1	1	yes	characteristic
66	#10596831	—	2	4	no	—
67	#10596831	—	4	2	no	—
68	C₂²×C₇₃	—	2	1	no	—
69	C₂²×C₇₃	—	2	1	no	—
70	C₂²×C₇₃	—	2	1	no	—
71	C₂²×C₇₃	#336	1	1	yes	characteristic
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	—	2	1	no	—
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	#343	1	1	yes	characteristic
76	C₄×C₇₃	#344	1	1	yes	characteristic
77	C₄×C₇₃	#350	1	2	yes	—
78	C₄×C₇₃	#364	1	2	yes	—
79	#7212	—	73	1	no	maximal, Sylow
80	C₂×C₇₃	—	4	1	no	—
81	C₂×C₇₃	#1283	1	1	yes	characteristic
82	C₂×C₇₃	#1376	1	1	yes	characteristic
83	C₂×C₇₃	#1379	1	1	yes	characteristic
84	#2162	—	73	1	no	—
85	#2150	D₇₃	1	1	yes	characteristic
86	C₈×M₁₆	—	73	1	no	—
87	C₇₃	#7212	1	1	yes	characteristic, complemented, Sylow
88	C₂×M₃₂	Dic₇₃	1	2	yes	—
89	#431	—	73	1	no	—
90	C₄×M₁₆	—	73	1	no	—
91	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
92	C₄⋊C₁₆	—	2	1	no	—
93	C₄⋊C₁₆	—	2	1	no	—
94	C₄⋊C₁₆	—	146	1	no	—
95	C₂²⋊C₁₆	—	73	2	no	—
96	C₄×C₁₆	—	146	1	no	—
97	#322	—	146	1	no	—
98	C₈²	—	146	1	no	—
99	C₂×M₁₆	—	146	1	no	—
100	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
101	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
102	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
103	M₃₂	—	4	2	no	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	—	146	2	no	—
106	C₄⋊C₈	—	73	2	no	—
107	C₂²⋊C₈	—	146	1	no	—
108	#97	—	73	2	no	—
109	C₄×C₈	—	146	1	no	—
110	C₄×C₈	—	146	1	no	—
111	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
112	C₄×C₈	D₂₉₂	1	1	yes	characteristic
113	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
114	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
115	C₂²×C₄	C₄×Dic₇₃	1	1	yes	characteristic
116	C₂²×C₄	#11924011	1	1	yes	characteristic
117	C₂²×C₄	#11924011	1	1	yes	characteristic
118	M₁₆	—	146	4	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	—	146	2	no	—
123	C₂×C₈	—	146	2	no	—
124	C₂×C₈	#11924005	1	1	yes	characteristic
125	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
126	C₂×C₈	#11924007	1	1	yes	characteristic
127	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
128	C₄²	#11924005	1	1	yes	characteristic
129	C₄²	#11924007	1	1	yes	characteristic
130	C₄²	C₈×D₇₃	1	1	yes	characteristic
131	C₄²	C₈×D₇₃	1	1	yes	characteristic
132	C₁₆	—	4	2	no	—
133	C₁₆	—	292	2	no	—
134	C₂³	#14556501	1	1	yes	characteristic
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	#14556498	1	1	yes	characteristic
140	C₂×C₄	C₈×Dic₇₃	1	1	yes	characteristic
141	C₂×C₄	#14556503	1	1	yes	characteristic
142	C₂×C₄	#14556503	1	1	yes	characteristic
143	C₂×C₄	#14556520	1	1	yes	characteristic
144	C₂×C₄	#14556520	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₈	—	292	2	no	—
149	C₈	—	146	4	no	—
150	C₈	#14556508	1	1	yes	characteristic
151	C₈	#14556510	1	1	yes	characteristic
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	#17988341	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	#17988356	1	1	yes	characteristic
160	C₄	#17988358	1	1	yes	characteristic
161	C₄	#17988372	1	2	yes	—
162	C₄	#17988407	1	2	yes	—
163	C₂	—	4	1	no	—
164	C₂	#22607781	1	1	yes	characteristic
165	C₂	#22608012	1	1	yes	characteristic
166	C₂	#22608021	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22611730

C₂

yes

maximal, characteristic

C₈×#11924012

C₂

yes

maximal, characteristic

C₇₃×#2150

C₂

yes

maximal, characteristic

#17988640

C₄

yes

characteristic

C₄×#11924012

C₄

yes

characteristic

C₄⋊#11924003

—

C₂²⋊#11924003

C₄

yes

—

C₄×#11924003

—

#17988309

—

C₈×#10596831

—

C₂×M₃₂×C₇₃

C₄

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₃

—

C₄⋊C₁₆×C₇₃

—

C₂×#11924012

—

C₂×#11924003

—

C₄⋊#10596831

C₈

yes

—

C₂²⋊#10596831

—

#14556502

C₈

yes

—

C₄×#10596831

—

C₄×#10596831

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

Q₈

yes

—

—

—

—

C₂²×C₄×C₇₃

C₄²

yes

characteristic

C₂²×C₄×C₇₃

M₁₆

yes

characteristic

C₂²×C₄×C₇₃

M₁₆

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

C₂²⋊C₄

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

#97

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#95

yes

characteristic

C₂×C₄×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₈

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#336

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₄×C₇₃

#364

yes

—

#7212

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

yes

characteristic

C₈×M₁₆

—

C₇₃

#7212

yes

characteristic, complemented, Sylow

C₂×M₃₂

Dic₇₃

yes

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

146

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

146

—

#322

—

146

—

C₈²

—

146

—

C₂×M₁₆

—

146

—

100

C₂²×C₈

C₄×D₇₃

yes

characteristic

101

C₂²×C₈

C₂×Dic₇₃

yes

characteristic

102

C₂×C₄²

C₄×D₇₃

yes

characteristic

103

M₃₂

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

146

—

106

C₄⋊C₈

—

107

C₂²⋊C₈

—

146

—

108

#97

—

109

C₄×C₈

—

146

—

110

C₄×C₈

—

146

—

111

C₄×C₈

Dic₁₄₆

yes

characteristic

112

C₄×C₈

D₂₉₂

yes

characteristic

113

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

114

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

115

C₂²×C₄

C₄×Dic₇₃

yes

characteristic

116

C₂²×C₄

#11924011

yes

characteristic

117

C₂²×C₄

#11924011

yes

characteristic

118

M₁₆

—

146

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

146

—

123

C₂×C₈

—

146

—

124

C₂×C₈

#11924005

yes

characteristic

125

C₂×C₈

C₂²⋊Dic₇₃

yes

characteristic

126

C₂×C₈

#11924007

yes

characteristic

127

C₂×C₈

C₄⋊Dic₇₃

yes

characteristic

128

C₄²

#11924005

yes

characteristic

129

C₄²

#11924007

yes

characteristic

130

C₄²

C₈×D₇₃

yes

characteristic

131

C₄²

C₈×D₇₃

yes

characteristic

132

C₁₆

—

133

C₁₆

—

292

—

134

C₂³

#14556501

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

#14556498

yes

characteristic

140

C₂×C₄

C₈×Dic₇₃

yes

characteristic

141

C₂×C₄

#14556503

yes

characteristic

142

C₂×C₄

#14556503

yes

characteristic

143

C₂×C₄

#14556520

yes

characteristic

144

C₂×C₄

#14556520

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

292

—

149

C₈

—

146

—

150

C₈

#14556508

yes

characteristic

151

C₈

#14556510

yes

characteristic

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

#17988341

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

#17988356

yes

characteristic

160

C₄

#17988358

yes

characteristic

161

C₄

#17988372

yes

—

162

C₄

#17988407

yes

—

163

C₂

—

164

C₂

#22607781

yes

characteristic

165

C₂

#22608012

yes

characteristic

166

C₂

#22608021

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₇₃
commutator	C₂×C₄×C₇₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₃
Fitting	C₇₃×#2150	C₂
socle	C₂²×C₇₃	#336
prime radical	C₂³×C₇₃	#97
cosocle	C₂×C₄×C₈×C₇₃	C₂²
hypercenter	#2150	D₇₃
hypocenter	C₇₃	#7212
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂²⋊Dic₇₃

commutator

C₂×C₄×C₇₃

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₇₃

Fitting

C₇₃×#2150

C₂

C₂²×C₇₃

C₂³×C₇₃

C₂×C₄×C₈×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
87. C₇₃	#7212	semidirect

H ≅ G/N

type

87. C₇₃

#7212

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22611730	2
C₈×#11924012	2
C₇₃×#2150	2
#7212	73

subgroup

index

#22611730

C₈×#11924012

C₇₃×#2150

#7212

Subgroups — #32973006

Conjugacy classes of subgroups

Complements

Maximal subgroups